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  "body": "\n## Statement\n\nLet $G$ be a [[finitely generated group|082DWWXMRJ]].\nThen $G$ has polynomial [[growth|4CMRM3975W]] if and only if $G$ is virtually nilpotent, that is, $G$ has a [[nilpotent|0Q7MMQHC1P]] [[subgroup|0X1JXT47GA]] of finite [[index|0DTXBQPA2S]].\n\n## Remarks\n\nThe direction \"virtually nilpotent implies polynomial growth\" was known before Gromov.\nFor a finitely generated nilpotent group the degree of growth is the integer $\\sum_{k \\ge 1} k \\, r_k$.\nHere $r_k$ is the torsion-free rank of the quotient $\\gamma_k(G) / \\gamma_{k+1}(G)$ of consecutive terms of the lower central series.\nGromov proved the hard direction.\nHis idea is to rescale the word metric of $G$ by $1/n$ and pass to a limit space as $n \\to \\infty$.\nPolynomial growth makes this limit a finite-dimensional, locally compact space.\nThe solution of Hilbert's fifth problem turns its isometry group into a Lie group, on which $G$ acts.\nFrom this action one extracts the nilpotent subgroup by induction on the degree.\nKleiner later gave a different proof based on harmonic functions on the Cayley graph.\n\nExample: the integer Heisenberg group of upper unitriangular $3 \\times 3$ integer matrices is nilpotent, with $r_1 = 2$ and $r_2 = 1$, so its growth is polynomial of degree $2 + 2 = 4$.\n\nA consequence is a gap: the growth of a finitely generated group is either polynomial of an integer degree, or faster than every polynomial.\nGroups of intermediate growth exist, such as the [[Grigorchuk group|M0WAT1VQZH]], so \"faster than every polynomial\" does not mean exponential.\n\n## Explanations\n\n### L1 human\n\nCount the elements of a group that can be reached in at most $n$ steps from the identity.\nIf this count grows only like a power of $n$, the theorem says the group is very close to being commutative: up to a finite index, it is nilpotent.\nSo a purely geometric counting condition forces a strong algebraic structure.\n\n### L2 human\n\nThe theorem shows that being virtually nilpotent is a [[quasi-isometry|6FKYSH729T]] invariant of finitely generated groups, because growth is.\nFor finitely generated linear groups the gap is sharper: by the [[Tits alternative|QYXGVES1XM]] and the Milnor–Wolf theorem for solvable groups, their growth is either polynomial or exponential.\nIt is also used in the study of random walks and of manifolds with almost nonnegative curvature.\n",
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